Abstract Diagrams and an Opfibration Account of Typed Graph Transformation

Diagrams and an Opfibration Account of Typed Graph Transformation R. Banach Computer Science Dept., Manchester University, Manchester, M13 9PL, U.K. [email protected] A. Corradini Dipartimento di Informatica, Università di Pisa, Corso Italia 40, Pisa, Italy. [email protected] Abstract: The “in the large” properties of typed graph transformation systems in the double pushout framework and a double pullback variation of it, are reexamined. Preceding accounts utilising a fixed choice of pullbacks (whether adopted directly or via partial morphisms) are seen to be excessively sensitive to the precise graphs involved for comfort. A theory of abstract diagrams is developed, that allows the smooth formulation of an abstract version of the theory. Graph transformation steps appear as a split opfibration over abstract type change. The category of graph grammars, the category of graph transition systems, and the category of graph derivation systems emerge as opfibrations over abstract type change. Weakening the level of abstraction to the extent used to preserve event identity in event based treatments of graph transformation phenomena, makes the transformation steps opfibration unsplit, and weakens certain adjunctions. All the properties of interest are combined in a single triple category. The “in the large” properties of typed graph transformation systems in the double pushout framework and a double pullback variation of it, are reexamined. Preceding accounts utilising a fixed choice of pullbacks (whether adopted directly or via partial morphisms) are seen to be excessively sensitive to the precise graphs involved for comfort. A theory of abstract diagrams is developed, that allows the smooth formulation of an abstract version of the theory. Graph transformation steps appear as a split opfibration over abstract type change. The category of graph grammars, the category of graph transition systems, and the category of graph derivation systems emerge as opfibrations over abstract type change. Weakening the level of abstraction to the extent used to preserve event identity in event based treatments of graph transformation phenomena, makes the transformation steps opfibration unsplit, and weakens certain adjunctions. All the properties of interest are combined in a single triple category.